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the deepest bonding MO (no nodal planes);
!
<
C
X
4
x
m
sinm
2
5
¼
x
1
sin
2
5
þx
2
sin
4
5
þx
3
sin
6
5
þx
4
sin
8
5
f
2
¼
C
m
¼
1
:
¼
ð
:
9510
x
1
þ
:
5878
x
2
:
5878
x
3
:
9510
x
4
Þ
C 0
0
0
0
¼
0
:
6015
x
1
þ
0
:
3718
x
2
0
:
3718
x
3
0
:
6015
x
4
ð
3
:
20
Þ
the second bonding MO (HOMO, one nodal plane);
8
<
:
!
C
X
4
x
m
sinm
3
5
¼
x
1
sin
3
5
þx
2
sin
6
5
þx
3
sin
9
5
þx
4
sin
12
p
f
3
¼
C
5
m
¼
1
¼
C 0
ð
:
9510
x
1
0
:
5878
x
2
0
:
5878
x
3
þ
0
:
9510
x
4
Þ
¼
0
:
6015
x
1
0
:
3718
x
2
0
:
3718
x
3
þ
0
:
6015
x
4
ð
3
:
21
Þ
the first antibonding MO (LUMO, two nodal planes);
0
@
1
A
<
:
C
X
4
x
m
sinm
4
5
¼
x
1
sin
4
5
þx
2
sin
8
5
þx
3
sin
12
5
þx
4
sin
16
5
f
4
¼
C
m
¼
1
¼
C 0
ð
:
5878
x
1
0
:
9510
x
2
þ
0
:
9510
x
3
0
:
5878
x
4
Þ
¼
0
:
3718
x
1
0
:
6015
x
2
þ
0
:
6015
x
3
0
:
3718
x
4
ð
3
:
22
Þ
the last antibonding MO, highest in energy (three nodal planes). These
MOs coincide with those given in Equations (2.272) of Chapter 2, and
whose shapes are sketched in Figure 2.27.
3.2 THE CLOSED POLYENE CHAIN
Next, we want to find the general solution for the systemof homogeneous
linear equations for the closed polyene chain with N atoms yielding the
Nth degree secular equation
